Prop

Every circle 2-bundle/bundle gerbe on XX is equivalent to the lifting gerbe of some PU(ℋ)P U(\mathcal{H})-principal bundle to a U(ℋ)U(\mathcal{H})-bundle, and the equivalence classes of these structures correspond uniquely.

Action on Fredholm operators

Let ℋ\mathcal{H} be an infinite-dimensional separable Hilbert space.

Since by the above PU(ℋ)≃BU(1)PU(\mathcal{H}) \simeq B U(1) and since there is a canonical action of line bundles on complex vector bundles, hence on the topological K-theory of a manifold XX, there must also be a natural action of PU(ℋ)×Fred→FredPU(\mathcal{H}) \times Fred \to Fred of PU(ℋ)PU(\mathcal{H}) on the space of Fredholm operators (on connected components).

This is given by letting a projective unitary act by conjugation on a Fredholm operator: (g,F)↦gFg−1(g, F) \mapsto g F g^{-1}.